The R package Conics
Version 0.3
Bernard Desgraupes
University Paris Ouest
Modal’X
Last update: 2013-11-16
Contents
1 Algebraic background
1.1 Notation . . . . . . .
1.2 Classification . . . .
1.3 Points at infinity . .
1.4 Center . . . . . . . .
1.5 Axes . . . . . . . . .
1.6 Reduced equation . .
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2
2
2
3
4
4
5
2 The
2.1
2.2
2.3
2.4
2.5
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5
5
6
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13
conics package
Basic functions . . .
Plotting parameters
Aspect ratio . . . . .
Examples . . . . . .
Return value . . . .
1
The conics package - November 22, 2013
1
2
Algebraic background
This section is a survey of the main results concerning the algebraic representation of a plane conic.
1.1
Notation
A conic C is a plane algebraic curve of degree 2. It is the set of zeroes of a
polynomial of degree 2 in 2 variables, that is to say the set of points (x1 , x2 )
satisfying an equation of the form
P (x1 , x2 ) = a1 x21 + a2 x1 x2 + a3 x22 + a4 x1 + a5 x2 + a6 = 0.
(1)
Consider the following change of variables which introduces homogeneous
coordinates (X1 , X2 , X3 ):
x1 =
X1
X3
x2 =
X2
X3
(2)
Provided X3 6= 0, the previous equation can be simplified :
Q(X1 , X2 , X3 ) = a1 X12 +a2 X1 X2 +a3 X22 +a4 X1 X3 +a5 X2 X3 +a6 X32 = 0. (3)
It appears that Q(X1 , X2 , X3 ) is quadratic form. This form can be represented by a 3 × 3 matrix A such that
Q(X) = tXAX
The matrix is defined like this :

a
 1

1
A=
 2 a2

1
a4
2
1
a2
2
a3
1
a5
2

1
a4
2 

1 
a5 
2 


a6
(4)
(5)
It is a symmetric matrix. Let ∆ = det(A). If ∆ 6= 0, the conic is said to be
regular (or non-degenerate), otherwise it is degenerate. The same terminology
applies to the quadratic form itself.
When a quadratic form is degenerate, it splits into the product of two polynomials of degree 1. Geometrically, it means that the conic is a pair of lines.
On the contrary, if the quadratic form is non-degenerate, the conic is an ellipse,
a hyperbola, or a parabola.
1.2
Classification
In order to decide which kind of conic is represented by the matrix A, one must
consider the 2 × 2 top left submatrix, i-e the matrix B obtained by deleting the
last row and the last column of A:

1 
a1
a2
2 

B=
(6)

1
a2 a3
2
The conics package - November 22, 2013
3
The determinant of B is denoted δ. Its value is
1
δ = a1 a3 − a22 .
4
(7)
In the non-degenerate case, the matrix A has rank 3 and one has the following
classification based on the value of δ:
• if δ > 0, C is an ellipse
• if δ = 0, C is a parabola
• if δ < 0, C is a hyperbola
If the conic is degenerate, A has rank less than 3 and one has the following
classification:
• if δ > 0, C is empty
• if δ = 0, C is a pair of parallel lines (possibly coincident)
• if δ < 0, C is a pair of intersecting lines
In particular, the case of a double line (coincident parallel lines) occurs when A
is of rank 1.
1.3
Points at infinity
Except in the case of an ellipse, all the conics have points at infinity. These
points can be found by letting X3 −→ 0 in equation (3). One obtains the
following equation:
a1 X12 + a2 X1 X2 + a3 X22 = 0
which can be rewritten in variables x1 and x2 like this
a1 x21 + a2 x1 x2 + a3 x22 = 0.
(8)
x2
. The variable t can be interpreted as the slope of the directions
x1
to infinity. The previous equation becomes, after division by x21 :
Let t =
a1 + a2 t + a3 t2 = 0
(9)
It is an ordinary equation of degree 2 which will have real solutions if its
discriminant is non-negative :
D = a22 − 4a1 a3 = −4δ ≥ 0
(10)
So, if δ > 0 (case of an ellipse), the discriminant is negative and there are
no solutions: this is normal since an ellipse does not have points at infinity. If
δ < 0 (case of a hyperbola), one finds two distinct solutions which correspond
to the slope of the asymptotes of the hyperbola. Finally, if δ = 0 (case of a
parabola), one finds a unique solution which is the asymptotic direction of the
branches of the parabola.
The conics package - November 22, 2013
1.4
4
Center
Some conics have a center C. In the center, the gradient of the quadratic
polynomial P is null. This leads to the following equations:

∂P


=0


 ∂x1
(11)


∂P



=0
∂x2
The partial derivatives yield the following equations:
1
a1 x1 + a2 x2 + a4
2

1
 a x + 1a x + a
2 1
2 3
5
2
2



=
0
(12)
=
0
This is a system of two linear equations in two variables. Its matrix is B.
If δ = det(B) 6= 0, it has a unique solution and the conic has a unique center.
This is the case of an ellipse, or a hyperbola or a pair of intersecting lines.
1.5
Axes
The symmetry axes of a conic are lines passing through the center. Their direction vectors are the eigenvectors of the submatrix B defined by (6).
Since B is symmetric, one has the following properties:
• the eigenvalues λ1 and λ2 are real (not complex);
• the eigenvectors are real too;
• the matrix can always be diagonalized in an orthonormal basis. It means
that one can always find two orthogonal eigenvectors with norm equal to
1. Let us denote V1 and V2 these two vectors.
As a consequence, a conic has in general two axes which are orthogonal.
The eigenvalues are the roots of the characteristic polynomial associated to
matrix B. It is defined as
P (λ) = det(B − λI)
= λ2 − Tr(B)λ + det(B)
= λ2 − (a1 + a3 )λ + δ
(13)
=0
If λ is an eigenvalue, the corresponding eigenvector V can be calculated by
solving the following equation :
(B − λI)V = 0
(14)
In the particular case where λ1 = λ2 , the eigenspace has dimension 2 which
means that any direction is a possible eigenvector. This corresponds to a circle:
in a circle indeed any diameter is a symmetry axis.
The conics package - November 22, 2013
1.6
5
Reduced equation
In the case of a conic with a center (ellipse or hyperbola), one can change the
coordinate system by translating the origin to the center C and by rotating
the axes to vectors V1 and V2 . In the {C, V1 , V2 } basis, let us designate the
coordinates by y1 and y2 . The equation of the conic in this basis is remarkably
simple :
∆
λ1 y12 + λ2 y22 +
=0
(15)
δ
The relation between the (x1 , x2 ) and (y1 , y2 ) coordinates are given by the
transformation matrix T whose columns are the eigenvectors V1 and V2 . One
has
!
!
x1
y1
=T
.
(16)
x2
y2
2
The conics package
The R package conics makes use of the previous results to plot conic curves.
The package must be loaded with the library command like this:
> library(conics)
2.1
Basic functions
In the R conics package, conics can be specified either by a 6-length vector
containing the coefficients of the polynomial in equation (1), or by the symmetric
matrix A defined by equation (5).
There is a convenience function named conicMatrix which computes the
matrix A given the vector of coefficients of the polynomial P defined by (1).
Here is a simple example: let us consider the conic with equation
2 x21 + 2 x1 x2 + 2 x22 − 20 x1 − 28 x2 + 10 = 0
The vector of coefficients is
> v <- c(2,2,2,-20,-28,10)
and the corresponding matrix can be obtained by the following instruction:
> A <- conicMatrix(v)
[1,]
[2,]
[3,]
[,1] [,2] [,3]
2
1 -10
1
2 -14
-10 -14
10
The center and the axes of the conic can be calculated using the functions
conicCenter and conicAxes respectively. For instance:
> conicCenter(v)
[1] 2 6
The conics package - November 22, 2013
6
> conicAxes(v)
[,1]
[,2]
[1,] 0.7071068 -0.7071068
[2,] 0.7071068 0.7071068
Alternatively, one can specify the matrix instead of the vector:
> conicCenter(A)
> conicAxes(A)
Finally the conic can be plotted with the conicPlot function like this:
> conicPlot(v, main="conicPlot(v)", xlab="", ylab="")
0
5
10
conicPlot(v)
−5
2.2
0
5
10
Plotting parameters
The conicPlot function calculates a set of points on the conic and ultimately
calls the usual plot function from the graphics package. Any of the numerous
arguments defined with the plot function can be specified in the conicPlot
function as well. For instance, in order to draw the previous ellipses in red with
a dotted contour, one can write :
conicPlot(v, col="red", lty=3)
The conicPlot function also has a set of optional arguments of its own.
Currently the following arguments are defined :
The conics package - November 22, 2013
7
add is a boolean argument. If it is set to TRUE, the drawing is added to the
current plot instead of erasing the current graphical device.
as.col specifies the color of the asymptotes. It can take the same values as the
col argument of the plot function.
as.lty specifies the line type for the asymptotes. It can take the same values as
the lty argument of the plot function.
ax.col specifies the color of the axes. It can take the same values as the col
argument of the plot function.
ax.lty specifies the line type for the axes. It can take the same values as the
lty argument of the plot function.
asymptotes is a boolean argument. If it is set to TRUE, the asymptotes will be
drawn. This argument is meaningful only in the case of a hyperbola.
center is a boolean argument. If it is set to TRUE, the center will be marked
by a small circle.
npoints is a numeric argument indicating the number of points to calculate in
order to draw the curve. The default value is 100.
sym.axes is a boolean argument. If it is set to TRUE, the symmetry axes will
be drawn.
... any other arguments will be passed verbatim to the basic plot function from
the graphics package. See the documentation of the par function to know
which arguments are supported. In particular, the following two arguments are very useful:
xlim is a 2-elements numeric vector specifying the range of the x-coordinate.
ylim is a 2-elements numeric vector specifying the range of the y-coordinate.
2.3
Aspect ratio
In order to avoid distorsions due to the difference of units between the x-axis
and the y-axis, the asp argument can be very useful. It is defined as the ratio
in length between one data unit in the y direction and one data unit in the x
direction.
Setting asp = 1 will ensure that the same unit length is used for both
coordinate axes so that distances between points are represented accurately.
For instance:
conicPlot(v, asp=1)
The conics package - November 22, 2013
2.4
8
Examples
Here is an example using the previous vector and demonstrating the center and
the sym.axes parameters :
10
> conicPlot(v, center=T, sym.axes=T)
0
5
y
●
−5
0
5
x
10
The conics package - November 22, 2013
9
Here is another example where several ellipses are drawn on the same plot
using the add parameter :
v <- c(2,2,2,-20,-28,10)
conicPlot(v, center=T, lty=1)
v[6] <- 30
conicPlot(v, add=T, lty=2)
v[6] <- 50
conicPlot(v, add=T, lty=4)
10
>
>
>
>
>
>
0
5
y
●
−5
0
5
x
10
The conics package - November 22, 2013
10
Here is now an example of a hyperbola making use of the xlim and ylim
parameters :
−5
0
5
y1
10
15
> v <- c(2,2,-2,-20,20,10)
> conicPlot(v, xlim=c(-10,10), ylim=c(-5,18))
−10
−5
0
x1
5
10
The conics package - November 22, 2013
11
Here is an example with the same hyperbola demonstrating the as.col, as.lty,
ax.col, and ax.lty options :
−5
0
5
y1
10
15
> conicPlot(v, asymptotes=T, sym.axes=T,
+ as.col="red", as.lty=2, ax.col="blue", ax.lty=4,
+ xlim=c(-10,10), ylim=c(-5,18))
−10
−5
0
x1
5
10
The conics package - November 22, 2013
12
Here is an example of extra arguments which are ultimately passed to the
plot function :
> conicPlot(v, asymptotes=T, sym.axes=T,
+ xlim=c(-10,10), ylim=c(-5,18),
+ asp=1, col="blue", main="Hyperbola", bty="n")
−5
0
5
y1
10
15
Hyperbola
−10
−5
0
5
10
x1
The asp argument (aspect ratio) is set to 1 to ensure accurate distances
between points. The col argument sets the color of the conic itself. The main
argument adds a title to the plot. The bty argument set to ”n” suppresses the
box around the plot.
The conics package - November 22, 2013
2.5
13
Return value
The return value of the conicPlot function is invisible, i-e it is not printed
on the console by default but it can be stored in a variable in order to get its
contents.
It is a list of computed values corresponding to various elements of the conic.
Some of the following elements can be found in the return list, depending on
the kind of the conic:
kind the kind of the conic. It is a character string whose value is ”ellipse”,
”hyperbola”, ”parabola”, or ”lines” ;
axes the symmetry axes. This value is a 2 × 2 matrix whose columns are
direction vectors of the axes ;
center the center of the conic ;
asymptotes the slopes of the asymptotes ;
vertices the vertices of the conic ;
foci the focal points of the conic ;
eccentricity the eccentricity of the conic. It is a value between 0 and 1 for an
ellipse, equal to 1 for a parabola and greater than 1 for an hyperbola ;
intercepts the intercepts in the case of parallel lines.
points the coordinates of the points used to plot the conic. This value is
returned only if the type option is equal to ‘n’ and if the conic is nondegenerate. In that case, nothing is drawn.
Here is an example
> v <- c(2,2,-2,-20,20,10)
[1]
2
2
-2 -20
20
10
> res <- conicPlot(v)
$kind
[1] "hyperbola"
$axes
[,1]
[,2]
[1,] -0.9732490 0.2297529
[2,] -0.2297529 -0.9732490
$center
[1] 2 6
$asymptotes
[1] 1.618034 -0.618034
$vertices
The conics package - November 22, 2013
14
$vertices$x
[1] 3.0864345 0.9135655
$vertices$y
[1] 1.39779 10.60221
$foci
$foci$x
[1] 3.5364504 0.4635496
$foci$y
[1] -0.5085083 12.5085083
$eccentricity
[1] 1.414214
In the next graphic, the vertices and the foci are drawn like this :
v <- c(-4,0,1,0,0,1)
cp <- conicPlot(v, sym.axes=TRUE, asymptote=TRUE, asp=1,
ax.lty=2, as.col="gray")
points(cp$foci$x,cp$foci$y,col="red",pch=19)
text(cp$foci$x,cp$foci$y+0.1,paste("F",2:1,sep=""))
points(cp$vertices$x,cp$vertices$y,col="blue",pch=19)
0.0
F1
F2
●
●
●
−0.5
●
−1.0
y1
0.5
1.0
>
>
+
>
>
>
−1.5
−1.0
−0.5
0.0
x1
0.5
1.0
1.5